Maximum likelihood estimation

Definition

Maximum likelihood estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. This technique provides a way to infer values for unknown parameters based on observed data, making it particularly valuable in various contexts such as probability distributions and statistical inference.

5 Must Know Facts For Your Next Test

1. In maximum likelihood estimation, the 'maximum' refers to the goal of finding the parameter values that make the observed data most likely.
2. MLE can be applied to various types of distributions, including normal, binomial, and Poisson distributions.
3. One key feature of MLE is that it produces estimators that are consistent, meaning they converge in probability to the true parameter value as the sample size increases.
4. MLE estimators have desirable properties such as asymptotic normality, which allows for the construction of confidence intervals and hypothesis tests.
5. The method can sometimes lead to biased estimates in small samples but tends to perform well as sample size increases.

Review Questions

• How does maximum likelihood estimation relate to the concept of likelihood functions and what role does it play in parameter estimation?
  • Maximum likelihood estimation relies on likelihood functions, which quantify how likely it is to observe the given data under various parameter values. By maximizing this likelihood function, MLE identifies the parameter values that best explain the observed data. This relationship emphasizes how MLE seeks to find parameter estimates that maximize the probability of obtaining the observed dataset within a specified statistical model.
• Discuss how maximum likelihood estimation can be utilized in conjunction with different probability distributions like Poisson and normal distributions.
  • Maximum likelihood estimation can be effectively used with various probability distributions, including Poisson and normal distributions, by deriving specific likelihood functions for each case. For instance, when estimating parameters for a Poisson distribution representing count data, MLE maximizes the likelihood function specific to that distribution. Similarly, for a normal distribution dealing with continuous data, MLE finds estimates for mean and variance by maximizing its respective likelihood function, showcasing its versatility across different types of data and models.
• Evaluate the implications of using maximum likelihood estimation in statistical inference, particularly concerning its properties and limitations in practical applications.
  • Using maximum likelihood estimation in statistical inference carries significant implications regarding its properties and limitations. While MLE provides consistent and asymptotically normal estimators, ensuring reliable inference as sample sizes increase, it may yield biased results in smaller samples. Additionally, while MLE can be powerful in practice, challenges such as model mis-specification or non-identifiability may arise, impacting the validity of estimates. Understanding these nuances is crucial for effectively applying MLE in real-world statistical analysis.

Related terms

Likelihood function: A function that measures the probability of observing the given data under different parameter values of a statistical model.

Statistical inference: The process of drawing conclusions about a population or process based on a sample of data and statistical models.

Estimation theory: A branch of statistics that deals with the estimation of parameters based on observed data, including methods like MLE and method of moments.